Trigonometrical Ratios Allied Angles (Negative and Associated Angles)

1. Compound angle formulae of two angles:

(i) $\sin \left(A+B\right)=\sin A\cos B+\cos A\sin B$

(ii) $\sin \left(A-B\right)=\sin A\cos B-\cos A\sin B$

(iii) $\cos \left(A+B\right)=\cos A\cos B-\sin A\sin B$

(iv) $\cos \left(A-B\right)=\cos A\cos B+\sin A\sin B$

(v) $\mathrm{t}\mathrm{a}\mathrm{n}\left(A+B\right)=\frac{\mathrm{t}\mathrm{a}\mathrm{n}A+\mathrm{t}\mathrm{a}\mathrm{n}B}{1-\mathrm{t}\mathrm{a}\mathrm{n}A\mathrm{t}\mathrm{a}\mathrm{n}B}$

(vi) $\mathrm{t}\mathrm{a}\mathrm{n}\left(A-B\right)=\frac{\mathrm{t}\mathrm{a}\mathrm{n}A-\mathrm{t}\mathrm{a}\mathrm{n}B}{1+\mathrm{t}\mathrm{a}\mathrm{n}A\mathrm{t}\mathrm{a}\mathrm{n}B}$

(vii) $\sin \left(A+B\right)\sin \left(A-B\right)={\sin }^{2}A-{\sin }^{2}B$

(viii) $\cos \left(A+B\right)\cos \left(A-B\right)={\cos }^{2}A-{\sin }^{2}B$

2. Compound angle formulae involving three angles:

(i) $\sin \left(A+B+C\right)=\sin A\cos B\cos C+\cos A\sin B \cos C+\cos A\cos B\sin C-\sin A\sin B\sin C$

(ii) $\cos \left(A+B+C\right)=\cos A\cos B\cos C-\cos A\sin B\sin C-\sin A\cos B\sin C-\sin A\sin B\cos C$

(iii) $\mathrm{t}\mathrm{a}\mathrm{n}\left(A+B+C\right)=\mathrm{ }\frac{\mathrm{t}\mathrm{a}\mathrm{n}A+\mathrm{t}\mathrm{a}\mathrm{n}B+\mathrm{t}\mathrm{a}\mathrm{n}C-\mathrm{t}\mathrm{a}\mathrm{n}A\mathrm{t}\mathrm{a}\mathrm{n}B\mathrm{t}\mathrm{a}\mathrm{n}C}{1-\mathrm{t}\mathrm{a}\mathrm{n}A\mathrm{t}\mathrm{a}\mathrm{n}B-\mathrm{t}\mathrm{a}\mathrm{n}B\mathrm{t}\mathrm{a}\mathrm{n}C-\mathrm{t}\mathrm{a}\mathrm{n}C\mathrm{t}\mathrm{a}\mathrm{n}A}$

3. Trigonometric identities when sum of angles is supplementary and complete:

(i) If $A+B=\mathrm{\pi },$ then $\sin A=\sin B,\cos A=-\cos B$ and $\tan A=-\tan B$

(ii) If $A+B=2\mathrm{\pi },$ then $\sin A=-\sin B,\cos A=\cos B$ and $\tan A=-\tan B$

4. Allied angles:

Two angles (or numbers) are called allied iff their sum or difference is a multiple of $\frac{\mathrm{\pi }}{2}$.

5. Rules to memorize the formulae of Allied angles:

(i) For trigonometric functions of (odd multiples of $\frac{\pi }{2}\pm \theta$) the trigonometric functions will interchange as $\sin \theta \leftrightarrow \cos \theta ,$ $\sec \theta \leftrightarrow cosec\theta ,$ $\tan \theta \leftrightarrow \cot \theta .$ Look at the quadrant to decide the sign of the trigonometric function.

(ii) For trigonometric functions of (multiple of$\pi \pm \theta$), there is no change in the trigonometric functions. Look at the quadrant to decide the sign of the trigonometric function.

6. Trigonometric functions of $\frac{\mathrm{\pi }}{2}-\theta$:

(i) $\sin \left(\frac{\pi }{2}-\theta \right)=\cos \theta$

(ii) $\cos \left(\frac{\pi }{2}-\theta \right)=\sin \theta$

(iii) $\tan \left(\frac{\pi }{2}-\theta \right)=\cot \theta$

(iv) $cosec\left(\frac{\pi }{2}-\theta \right)=\sec \theta$

(v) $\sec \left(\frac{\pi }{2}-\theta \right)=cosec\theta$

(vi) $\cot \left(\frac{\pi }{2}-\theta \right)=\tan \theta$

7. Trigonometric functions of $\frac{\mathrm{\pi }}{2}+\theta$:

(i) $\sin \left(\frac{\pi }{2}+\theta \right)=\cos \theta$

(ii) $\cos \left(\frac{\pi }{2}+\theta \right)=-\sin \theta$

(iii) $\tan \left(\frac{\pi }{2}+\theta \right)=-\cot \theta$

(iv) $cosec\left(\frac{\pi }{2}+\theta \right)=\sec \theta$

(v) $\sec \left(\frac{\pi }{2}+\theta \right)=-cosec\theta$

(vi) $\cot \left(\frac{\pi }{2}+\theta \right)=-\tan \theta$

8. Trigonometric identities from product to sum or difference:

(i) $2\sin A\cos B=\sin \left(A+B\right)+\sin \left(A-B\right)$

(ii) $2\cos A\sin B=\sin \left(A+B\right)-\sin \left(A-B\right)$

(iii) $2\cos A\cos B=\cos \left(A+B\right)+\cos \left(A-B\right)$

(iv) $2\sin A\sin B=\cos \left(A-B\right)-\cos \left(A+B\right)$

9. Trigonometric identities from sum or difference to product:

(i) sinC+sinD=2sinC+D2 cosC-D2

(ii) sinC-sinD=2sinC-D2cosC+D2

(iii) cosC+cosD=2cosC+D2cos C-D2

(iv) cosC-cosD=-2sinC+D2sinC-D2

10. Trigonometric identities of $2x$:

(i) $\sin 2x=2\sin x \cos x$

(ii) $\cos 2x={\cos }^{2}x-{\sin }^{2}x$

(iii) $\cos 2x=2{\cos }^{2}x-1$

(iv) $\cos 2x=1-2{\sin }^{2}x$

(v) $\mathrm{t}\mathrm{a}\mathrm{n}2x=\frac{2\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{ }x}{1-\mathrm{t}\mathrm{a}{\mathrm{n}}^2\mathrm{ }x}$

(vi) $\mathrm{s}\mathrm{i}\mathrm{n}2x=\frac{2\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{ }x}{1+\mathrm{t}\mathrm{a}{\mathrm{n}}^2\mathrm{ }x}$

(vii) $\mathrm{c}\mathrm{o}\mathrm{s}2x=\frac{1-\mathrm{t}\mathrm{a}{\mathrm{n}}^2\mathrm{ }x}{1+\mathrm{t}\mathrm{a}{\mathrm{n}}^2\mathrm{ }x}$

11. Trigonometric identities of $3x$:

(i) $\sin 3x=3\sin x-4{\sin }^{3}x$

(ii) $\cos 3x=4{\cos }^{3}x-3\cos x$

(iii) $\mathrm{t}\mathrm{a}\mathrm{n}3x=\frac{3\mathrm{t}\mathrm{a}\mathrm{n}x-\mathrm{t}\mathrm{a}{\mathrm{n}}^{3}x}{1-3\mathrm{t}\mathrm{a}{\mathrm{n}}^{2}x}$

12. Trigonometric identities of $\frac{x}{2}$:

(i) $\mathrm{s}\mathrm{i}\mathrm{n}\frac{x}{2}=\sqrt{\frac{1-\mathrm{c}\mathrm{o}\mathrm{s}x}{2}}$

(ii) $\mathrm{c}\mathrm{o}\mathrm{s}\frac{x}{2}=\sqrt{\frac{1+\mathrm{c}\mathrm{o}\mathrm{s}x}{2}}$

(iii) $\mathrm{t}\mathrm{a}\mathrm{n}\frac{x}{2}=\sqrt{\frac{1-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{ }x}{1+\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{ }x}}$

13. Some useful results:

(i) $\cos x\cos 2x\cos 2^{2}x\cos 2^{3}x....\cos 2^{n-1}x$ = sin 2nx2n sin x

(ii) $\mathrm{s}\mathrm{i}\mathrm{n}x\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\mathrm{\pi }}{3}\mathrm{ }-x\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\mathrm{\pi }}{3}\mathrm{ }+x\right)=\frac{1}{4}\mathrm{s}\mathrm{i}\mathrm{n}3x$

(iii) $\mathrm{c}\mathrm{o}\mathrm{s}x\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathrm{\pi }}{3}\mathrm{ }-x\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathrm{\pi }}{3}\mathrm{ }+x\right)=\frac{1}{4}\mathrm{c}\mathrm{o}\mathrm{s}3x$

14. Values of trigonometric functions for special angles:

$\begin{array}{|ccccccc|}\hline \mathit{\theta }& \mathbf{15}°& \mathbf{75}°& \mathbf{18}°& \mathbf{36}°& \mathbf{54}°& \mathbf{72}°\\ \sin \theta & \frac{\sqrt{3}-1}{2\sqrt{2}}& \frac{\sqrt{3}+1}{2\sqrt{2}}& \frac{\sqrt{5}-1}{4}& \frac{\sqrt{10-2\sqrt{5}}}{4}& \frac{\sqrt{5}+1}{4}& \frac{\sqrt{10+2\sqrt{5}}}{4}\\ \cos \theta & \frac{\sqrt{3}+1}{2\sqrt{2}}& \frac{\sqrt{3}-1}{2\sqrt{2}}& \frac{\sqrt{10+2\sqrt{5}}}{4}& \frac{\sqrt{5}+1}{4}& \frac{\sqrt{10-2\sqrt{5}}}{4}& \frac{\sqrt{5}-1}{4}\\ \hline\end{array}$